PRELIMINARY SOLUTIONS. 



these supplementary equations were submitted at this stage of the computation to a new 

 scrutiny, and a term introduced containing v 3 , the correction to the adopted value of &amp;gt;i 

 revolution of the micrometer-screw in the Santiago Circle. 



The &quot; Supplementary Equations&quot; thus assume the following form : 



27.000 x- 3.648 y-f 19.715 z- 1.000 t 2 + 1.725 V 3 4- 30.768 w 4- 1F.200 =0 



3.648x4- 49. 286 y 4- 0.118 z + 8.346 t 2 + 4.428 v 3 - 5.949 w -- 170&quot;.945 =0 

 19.715x4- 0.118 y4- 25.951 z- 6.669 t 2 4- 6.672 V 3 + 20.674 w - 19&quot;.874 = 



1.000x4- 8.346 y- 6.669 z -f 27.000 t 2 + 4.287 V 3 - 0.424 w - 26&quot;. 800 =0 

 1.725X+ 4.428 y 4- 6.672 z + 4.287 t 2 + 43.604 v 3 4- 0.969 w - 44&quot;. 221 =0 



30.768 x 5.949 y + 20.674 z - 0.424 t 2 4- 0.969 V 3 4- 35.383 w + 21&quot;. 904 = 



and from the fifth equation we obtain by substitution 



V 3 = 4- 0&quot;.3893 

 the substitution of which in the other equations reduces the group to the form : 



27. 000 x 3.648y + 19. 715 z 1.000 t 2 + 30.768 w 4- 11&quot;.872 =0 



3.648x4- 49. 286 y 4- 0.118 z + 8.346 t 2 - 5.949 w 169&quot;. 221 =0 

 19.715 x 4- 0.118 y + 25. 951 z- 6.669 t^ + 20.674 w - 17&quot;. 277 =0 



1.000x4- 8.346y 6.669 z 4- 27.000 ta - 0.424 w - 25&quot;. 132 = 

 30. 768 x- 5. 949 y-f 20.674 z- 0.424 t 2 + 35.383 w + 22&quot;.281 =0 



or, after introducing t = 1&quot;.097, to the finally adopted value for the supplementary equations 

 from the Santiago meridian circle : 



27.000 x- 3.648 y 4- 19.715 z + 30.768 w 4- 10&quot;.775 =0 



3.648 x 4- 49.286 y 4- 0.118 z - 5.949 w 160&quot;.065 = 



19.715x4- 0.118 y 4- 25.951 z 4- 20.674 w- 24&quot;.593 =0 



30.768 x 5.949 y + 20.674 + 35.383 w 4- 21&quot;. 816 =0 



The four series of normal equations, thus brought to their ultimate form, now follow, 

 together with the values of the unknown quantities which they afford prior to multiplication 

 by any factor dependent upon the instrument or the observer. The results of this solution 

 furnish an approximate criterion for the relative value of the several series of observations, and 

 we may hence deduce their respective weights. The groups of equations, when multiplied each 

 by its own weight and added anew, will furnish us results as trustworthy as the materials at 

 our disposal permit, and beyond which it would be needless to push the investigation. 



